Following numerous studies making use of the numerical methods described above for inertial and capillary phenomena (see [1]}, [2]}, [3]}), we employ a fully-structured and uniform Cartesian grid divided into \(54 \times 4 \times 4\)  sub-domains, further divided into \(64 \times 32 \times 32\)  cells per sub-domain, rendering a global resolution equal to \(3456 \times 128 \times 128\)  cells. On account of ensuring this resolution produces mesh-independent results, we have performed a mesh independence study in surfactant-free and surfactant-laden conditions, detailed in Appendix  along with other resolution considerations. The mesh size selected also allowed to capture accurately the thin liquid film surrounding the bubble, which, according to the works of [4]} and [5]}, requires a minimum of 5-10 computational cells covering its domain for a correct development of the liquid velocity profile. In our simulations, we have ensured the placement of at least 5-6 cells in the liquid film section for the cases that exhibit encapsulation (\(We \sim 30\) ), for which \(h(x)/D \sim 5^{-2}\)  (see figure 11 of [5]}). This condition is further achieved in the surfactant-laden simulations, which develop a thicker liquid film. We make the note, however, that the size of the entire domain, as well as the comprehensive sweep over dimensionless numbers presented, make it prohibitively expensive in terms of computation to achieve the refinement levels reported in works such as [7]}, [8]} to fully capture all regime transitions leading to the interfacial singularities of pinch-off.
